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If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, what is f'(1)?

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Question: If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, what is f\'(1)?

Options:

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Correct Answer: 0

Solution: 

f\'(x) = 4x^3 - 12x^2 + 12x - 4; f\'(1) = 0.

If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, what is f'(1)?

Practice Questions

Q1
If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, what is f'(1)?
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Questions & Step-by-Step Solutions

If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, what is f'(1)?
  • Step 1: Identify the function f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1.
  • Step 2: Find the derivative of the function, denoted as f'(x).
  • Step 3: Use the power rule to differentiate each term of f(x):
  • - The derivative of x^4 is 4x^3.
  • - The derivative of -4x^3 is -12x^2.
  • - The derivative of 6x^2 is 12x.
  • - The derivative of -4x is -4.
  • - The derivative of the constant 1 is 0.
  • Step 4: Combine the derivatives to get f'(x) = 4x^3 - 12x^2 + 12x - 4.
  • Step 5: Now, substitute x = 1 into the derivative f'(x).
  • Step 6: Calculate f'(1): f'(1) = 4(1)^3 - 12(1)^2 + 12(1) - 4.
  • Step 7: Simplify the expression: f'(1) = 4 - 12 + 12 - 4.
  • Step 8: Combine the numbers: f'(1) = 0.
  • Differentiation – The process of finding the derivative of a function.
  • Polynomial Functions – Understanding the behavior and properties of polynomial functions.
  • Evaluation of Derivatives – Calculating the value of the derivative at a specific point.
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